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I stumbled across this question about relation from Discrete Mathematics and Its Application (8th Ed.) by Kenneth Rosen as follow:

  1. Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where $(x, y) ∈ R$ if and only if
    c.) $x = y + 1 \; or \; x = y − 1$ (Answer from the book: symmetric only)

Based on my understanding on antisymmetric definition as follow:
$∀x∀y(((x, y) ∈ R ∧ (y, x) ∈ R) \to (x = y))$

Since there exists no $(x, y)$ and $(y, x)$ that can make the case true, isn't it antisymmetric in this case for the question?

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  • $\begingroup$ Hint: rephrase it as $|x-y|=1$. $\endgroup$
    – J.G.
    Oct 9, 2022 at 18:57

1 Answer 1

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The given condition is $x=y+1$ or $x=y-1$. So, it is not antisymmetric: $1\mathrel R2$ and $2\mathrel R1$, but $1\ne2$.

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  • $\begingroup$ I got the idea thanks to @J.G. hint at the comment. Much appreciated! $\endgroup$
    – EverDream
    Oct 10, 2022 at 2:40

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